bigint-gcd

Greater common divisor (gcd) of two BigInt values using Lehmer's GCD algorithm. See https://en.wikipedia.org/wiki/Greatest_common_divisor#Lehmer's_GCD_algorithm. On my tests it is faster than Euclidean algorithm starting from 80-bit integers.

A version 1.0.2 also has something similar to "Subquadratic GCD" (see https://gmplib.org/manual/Subquadratic-GCD ), which is faster for large bigints (> 65000 bits), it should has better time complexity in case the multiplication is subquadratic, which is true in Chrome 93.

Installation

$ npm install bigint-gcd

Usage

import gcd from './node_modules/bigint-gcd/gcd.js';

console.log(gcd(120n, 18n));

There is also an implementation of the Extended Euclidean algorithm, which is useful to find the multiplicative modular inverse:

console.log(gcd.gcdext(3n, 5n)); // [2n, -1n, 1n]

And "Half GCD" which is useful to do the Rational reconstruction: It returns the transformation matrix and the transformed values after applying about half of the Euclidean steps.

console.log(gcd.halfgcd(1000000n, 1234567n)); // [-16n, 13n, 21n, -17n, 49371n, 12361n]

Performance:

The benchmark (see benchmark.html) resutls under Opera 87:

| bit size | gcd | gcd Julia 1.7.3 | gcdext | gcdx Julia 1.7.3 | invmod Julia 1.7.3 | halfgcd | | ------------------ | ------------------- | ------------------ | ------------------ | ------------------ | ------------------ | ------------------ | | 64 | 0.000230ms | 0.000310ms | 0.000420ms | 0.002550ms | 0.002620ms | 0.002040ms | | 128 | 0.001280ms | 0.000500ms | 0.002040ms | 0.004260ms | 0.001510ms | 0.005290ms | | 256 | 0.002900ms | 0.001800ms | 0.004000ms | 0.002560ms | 0.002500ms | 0.008510ms | | 512 | 0.005550ms | 0.003110ms | 0.007810ms | 0.004210ms | 0.004210ms | 0.011050ms | | 1024 | 0.012450ms | 0.006350ms | 0.018070ms | 0.007930ms | 0.007930ms | 0.017460ms | | 2048 | 0.029540ms | 0.012450ms | 0.045170ms | 0.016360ms | 0.015380ms | 0.038820ms | | 4096 | 0.070800ms | 0.025390ms | 0.157230ms | 0.037110ms | 0.034180ms | 0.087400ms | | 8192 | 0.172850ms | 0.069340ms | 0.388670ms | 0.095700ms | 0.086910ms | 0.207030ms | | 16384 | 0.480470ms | 0.164060ms | 0.943360ms | 0.269530ms | 0.238280ms | 0.466800ms | | 32768 | 1.582030ms | 0.468750ms | 2.468750ms | 0.828120ms | 0.738280ms | 1.152340ms | | 65536 | 4.250000ms | 1.421880ms | 6.625000ms | 2.109380ms | 2.757810ms | 3.015630ms | | 131072 | 11.156250ms | 3.796880ms | 17.953130ms | 5.671880ms | 4.953120ms | 8.015630ms | | 262144 | 30.906250ms | 10.406250ms | 49.125000ms | 18.437500ms | 15.843750ms | 21.625000ms | | 524288 | 81.437500ms | 33.187500ms | 128.750000ms | 43.875000ms | 40.750000ms | 54.500000ms | | 1048576 | 199.250000ms | 75.375000ms | 325.125000ms | 124.750000ms | 107.625000ms | 135.000000ms | | 2097152 | 484.000000ms | 185.500000ms | 804.250000ms | 302.000000ms | 279.750000ms | 331.250000ms | | 4194304 | 1170.000000ms | 432.000000ms | 1936.500000ms | 762.500000ms | 687.500000ms | 793.000000ms | | 8388608 | 2828.000000ms | 1081.000000ms | 4747.000000ms | 1798.000000ms | 1696.000000ms | 1902.000000ms |

Benchmark:

import {default as LehmersGCD} from './gcd.js';

function EuclideanGCD(a, b) {
  while (b !== 0n) {
    const r = a % b;
    a = b;
    b = r;
  }
  return a;
}

function ctz4(n) {
  return 31 - Math.clz32(n & -n);
}
const BigIntCache = new Array(32).fill(0n).map((x, i) => BigInt(i));
function ctz1(bigint) {
  return BigIntCache[ctz4(Number(BigInt.asUintN(32, bigint)))];
}
function BinaryGCD(a, b) {
  if (a === 0n) {
    return b;
  }
  if (b === 0n) {
    return a;
  }
  const k = ctz1(a | b);
  a >>= k;
  b >>= k;
  while (b !== 0n) {
    b >>= ctz1(b);
    if (a > b) {
      const t = b;
      b = a;
      a = t;
    }
    b -= a;
  }
  return k === 0n ? a : a << k;
}

function FibonacciNumber(n) {
  console.assert(n > 0);
  var a = 0n;
  var b = 1n;
  for (var i = 1; i < n; i += 1) {
    var c = a + b;
    a = b;
    b = c;
  }
  return b;
}

function RandomBigInt(size) {
  if (size <= 32) {
    return BigInt(Math.floor(Math.random() * 2**size));
  }
  const q = Math.floor(size / 2);
  return (RandomBigInt(size - q) << BigInt(q)) | RandomBigInt(q);
}

function test(a, b, f) {
  const g = EuclideanGCD(a, b);
  const count = 100000;
  console.time();
  for (let i = 0; i < count; i++) {
    const I = BigInt(i);
    if (f(a * I, b * I) !== g * I) {
      throw new Error();
    }
  }
  console.timeEnd();
}

const a1 = RandomBigInt(128);
const b1 = RandomBigInt(128);

test(a1, b1, LehmersGCD);
// default: 426.200927734375 ms
test(a1, b1, EuclideanGCD);
// default: 1136.77294921875 ms
test(a1, b1, BinaryGCD);
// default: 1456.793212890625 ms

const a = FibonacciNumber(186n);
const b = FibonacciNumber(186n - 1n);

test(a, b, LehmersGCD);
// default: 459.796875 ms
test(a, b, EuclideanGCD);
// default: 2565.871826171875 ms
test(a, b, BinaryGCD);
// default: 1478.333984375 ms